Costruzione dei numeri naturali in insiemistica.
requires only a dozen or so levels of sets-of-sets-of-... -sets-of-naturals.
ix Numeri e misura; teoria dei numeri.
Arturo Magidin Nov 26 '17 at 4:06
You need to remember that "at the time" (when the idea of going through the "construction" first happened) there was a "crisis of foundations":
1) many things that had been taken for granted had turned out to be incorrect (such as the belief that the Dirichlet principle always held, or that functions had to be differentiable at "most" points), and that certain types of arguments had been shown to lead to contradictions. There was a conscious, deliberate attempt at trying to justify even things that people had taken for granted. So, like Gauss had "constructed" the complex numbers as points on the plane, people were trying to either establish other constructions: the real numbers, the rationals, the integers, and try to make sure that properties that had been merely assumed to hold could actually be proven to hold.
2) Or at least to pinpoint what assumptions were needed to ensure that such properties could be derived (e.g., Hilbert greatly expanded Euclid's axioms to help justify many properties "established" in the Elements, that simply did not follow from the axioms given).
3) This in parallel with the Hilbert programme to try to show that certain methods of proof did not introduce contradictions.
In particular, while you could axiomatize the integers and then derive the natural numbers, historically this is not what happened, but
Descartes, for instance, refers to "imaginary quantities" when talking about negative numbers.
So the integers are a very late addition to the mathematical toolkit. And so,
those who wanted to go back and make sure the foundations were correct started
with natural numbers and derived the integers.